Numerical Investigation of Copper-Water Nanofluid in a Parallel Plate Channel

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Numerical Investigation of Copper-Water Nanofluid in a

Parallel Plate Channel

Saeb Ragani Lamooki

Submitted to the

Institute of Graduate Studies and Research

in partial fulfillment of the requirements for the Degree of

Master of Science

in

Mechanical Engineering

Eastern Mediterranean University

June 2014

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Approval of the Institute of Graduate Studies and Research

Prof. Dr. Elvan Yılmaz Director

I certify that this thesis satisfies the requirements as a thesis for the degree of Master of Science in Mechanical Engineering.

Prof. Dr. Uğur Atikol

Chair, Department of Mechanical Engineering

We certify that we have read this thesis and that in our opinion it is fully adequate in scope and quality as a thesis for the degree of Master of Science in Mechanical Engineering.

Prof. Dr. İbrahim Sezai Supervisor

Examining Committee

1. Prof. Dr. Fuat Egelioğlu

2. Prof. Dr. İbrahim Sezai

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ABSTRACT

Heat transfer behavior of Cu-water nanofluid in a two dimensional (infinite depth) rectangular duct is studied numerically for laminar flow, where the nanofluid has been considered as a Newtonian fluid. The governing continuity, momentum, and energy equations are discretized using finite volume approach and solved using SIMPLE method. The viscosity and thermal conductivity of nanofluid are determined by models proposed by Brinkman and Patel et al. Study has been conducted for a wide range of Reynolds number from 10 to 1500, for solid volume fractions between 0% and 5%. Top and bottom walls are considered for two cases of constant temperature and constant wall heat flux, while results for both uniform and parabolic entrance velocities are considered in each case. It has been observed that the rate of heat transfer increases with increase in solid volume fraction as well as increase in flow rate. Besides, higher heat transfer is observed for uniform entrance velocity compared to channel with parabolic inlet velocity.

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ÖZ

Bakır-Su nano-sıvısının ısı transferi davranış ve iki boyutlu dikdörtgen kanal içerisinde sayısal ve nümerik olarak laminer akım için gözlemlenip incelenmiştir. Burada nano-sıvı bir Newton sıvısı olarak düşünülmüştür. Akımın sürekliliği ve istikrarı, ivme ve enerji eşitlikleri sonlu elemanlar yöntem ve analizi kullanılarak ayrıştırılmıştır ve “Simple” yöntemi kullanılarak çözümlendirilmiştir. Nano-sıvının akışkanlığı ve termal iletkenliği Brinkman ve Patel modelleri ile belirlenmiştir. Çalışma 10 Reynolds sayısından 1500 Reynolds sayısına kadar olmak üzere çok geniş Reynolds sayısı aralığında %0 dan %5‟e kadar olan katı hacim yüzdeleri için yapılıp sürdürülmüştür. Alt ve üst duvarlar sabit sıcaklıkta ve sabit duvar ısı akışında olmak üzere iki farklı durumda düşünülüp incelenirken hem düzgün hem de parabolik giriş hızları sonuçları her iki durumda da ayrı ayrı incelenmiştir. Sıvı içindeki katı hacim yüzdesi arttıkça ısı transferinin yükseldiği görülmüştür. Bunun yanında düzgün hız girişli akışın, parbolik hız girişli akıştan daha yüksek ısı transferi sağladığı gözlemlenmiştir.

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ACKNOWLEDGEMENT

It was impossible for me to finish my dissertation without the help of my supervisor.

I would like to express my sincerest gratitude to Professor Dr. Ibrahim Sezai for his great help and support.

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LIST OF TABLES

Table 3.1: Thermophysical properties of water and Cu at 0°C ((Jiji, 2006),

(Incropera, Lavine, & DeWitt, 2011)) ... 13 Table 4.1: Results of grid independence test, at Re=500, pure fluid ... 28 Table 4.2: Local wall Nusselt number for clear fluid, constant wall temperature, comparison with theoretical results ... 29 Table 4.3: Local wall Nusselt number for clear fluid, constant wall heat flux,

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LIST OF FIGURES

Figure 3.1: Channel Geometry ... 13 Figure 3.2: Nonstaggered grid arrangement ... 23 Figure 4.1: Distribution of local Nusselt number at hot wall for different solid

volume fractions, Re=100 ... 31 Figure 4.2: Distribution of local Nusselt number at hot wall for different solid

volume fractions, Re=1500 ... 32 Figure 4.3: Distribution of local Nusselt number at hot wall for different Reynolds numbers, ϕ=2.5%. ... 33 Figure 4.4: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different solid volume fractions, Re=100 ... 34 Figure 4.5: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different solid volume fractions, Re=1000 ... 35 Figure 4.6: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different Reynolds numbers, ϕ=2.5% ... 36 Figure 4.7: Distribution of local Nusselt number at hot wall for different solid

volume fractions, Re=100 ... 37 Figure 4.8: Distribution of local Nusselt number at hot wall for different solid

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Figure 4.11: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different solid volume fractions, Re=1000

... 41 Figure 4.12: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different Reynolds numbers, ϕ=2.5%. ... 42 Figure 4.13: Average Nusselt number at hot wall for different Re and ϕ for uniform (U) and parabolic (P) entrance velocities, constant wall temperature ... 43 Figure 4.14: uniform (U) and parabolic (P) entrance velocities, L=1m, 𝑇𝑤=constant

... 45 Figure 4.15: Average Nusselt number at hot wall for different Re and ϕ for uniform (U) and parabolic (P) entrance velocities, constant wall heat flux ... 46 Figure 4.16: uniform (U) and parabolic (P) entrance velocities, L=1m, 𝑞_𝑤=constant

... 47 Figure 4.17: Average Nusselt number at hot wall, different Re and ϕ, for constant wall temperature (cons. T) and constant wall heat flux (cons. q), channel with

uniform entrance velocity ... 49 Figure 4.18: walls at constant temperature and constant heat flux, uniform entrance velocity, L=1m ... 50 Figure 4.19: Average Nusselt number at hot wall, different Re and ϕ, for constant wall temperature (cons. T) and constant wall heat flux (cons. q), channel with

parabolic entrance velocity... 51 Figure 4.20: walls at constant temperature and constant heat flux, parabolic entrance velocity, L=1m ... 52 Figure 4.21: Local and average Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

𝑅𝑒_𝐷𝑒𝑃𝑟 for different Reynolds

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Figure 4.22: Local and average Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

numbers, parabolic entrance velocity for a pure fluid (ϕ=0.0%)... 53 Figure 4.23: Local Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

𝑅𝑒_𝐷𝑒𝑃𝑟 for uniform and parabolic entrance

velocities and different Reynolds numbers for a pure fluid (ϕ=0.0%) ... 54 Figure 4.24: Average Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

𝑅𝑒_𝐷𝑒𝑃𝑟 for uniform and parabolic

entrance velocities and different Reynolds numbers for a pure fluid (ϕ=0.0%) ... 55 Figure 4.25: Local (Loc) and average (Ave) Nusselt number ratios, vs 𝜉 = 𝑥/𝐷 𝑒

𝑅𝑒_𝐷𝑒𝑃𝑟 for

uniform entrance velocity and constant wall temperature for ϕ=5.0% ... 56 Figure 4.26: Local (Loc) and average (Ave) Nusselt number ratios, vs 𝜉 = 𝑥/𝐷 𝑒

parabolic entrance velocity and constant wall temperature for ϕ=5.0% ... 56 Figure 4.27: Local and average Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

numbers, uniform entrance velocity for a pure fluid (ϕ=0.0%) ... 57 Figure 4.28: Local and average Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

numbers, parabolic entrance velocity for a pure fluid (ϕ=0.0%)... 58 Figure 4.29: Local Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

𝑅𝑒_𝐷𝑒𝑃𝑟 for uniform and parabolic entrance

velocities and different Reynolds numbers for a pure fluid (ϕ=0.0%) ... 59 Figure 4.30: Average Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

𝑅𝑒_𝐷𝑒𝑃𝑟 for uniform and parabolic

entrance velocities and different Reynolds numbers for a pure fluid (ϕ=0.0%) ... 59 Figure 4.31: Local Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

𝑅𝑒_𝐷𝑒𝑃𝑟 for constant wall temperature (C.T)

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xii Figure 4.32: Average Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

𝑅𝑒_𝐷𝑒𝑃𝑟 for constant wall temperature

(C.T) and constant wall heat flux (C.q), uniform entrance velocity and different Reynolds numbers for a pure fluid (ϕ=0.0%) ... 60 Figure 4.33: Local (Loc) and average (Ave) Nusselt number ratios, vs 𝜉 = 𝑥/𝐷 𝑒

uniform entrance velocity and constant wall heat flux for ϕ=5.0% ... 61 Figure 4.34: Local (Loc) and average (Ave) Nusselt number ratios, vs 𝜉 = 𝑥/𝐷 𝑒

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LIST OF SYMBOLS

t Time, [s]

V

Velocity vector, [𝑚 𝑠 ]

u Velocity component in x direction, [𝑚 𝑠 ] v Velocity component in y direction, [𝑚 𝑠 ] P Pressure, [Pa] x Horizontal coordinate, [m] y Vertical coordinate, [m] T Temperature, [K] 𝐶_𝑝 Specific heat, [𝐽 𝑘𝑔 𝐾] k Thermal conductivity, [𝑊 𝑚 𝐾] c Empirical constant, 𝑐 = 3.60 × 104 Pe Peclet number

𝑑𝑝 Solid particles diameter, [m]

𝑑𝑓 Molecular size of the base fluid, 2 Å

𝑢_𝑝 Brownian motion velocity of the particles, [𝑚 𝑠 ] 𝐾_𝑏 Boltzmann constant, [𝐽 𝐾 ]

𝐿𝑟𝑒𝑓 Reference length, [m]

L Channel length, [m] H Channel width, [m]

𝑁𝑢𝑥 Local wall Nusselt number, 𝑁𝑢 =𝑕 𝐿_𝑘𝑟𝑒𝑓

𝑓

𝑁𝑢

Average wall Nusselt number, 𝑁𝑢 =1

𝐿 𝑁𝑢𝑥 𝑑𝑥 𝐿

0

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xiv 𝐴_𝑐 Channel cross section area, [𝑚2_]

𝑞𝑤 Wall heat flux, [𝑊 𝑚2]

Re Reynolds number, 𝑅𝑒 =𝜌𝑓 𝑈𝑟𝑒𝑓 𝐻

𝜇𝑓

Pr Prandtl number, 𝑃𝑟 =𝜇𝑓_𝑘𝐶𝑝 𝑓

𝑓

𝑈_𝑚 Mean inlet velocity, [𝑚 𝑠 ] 𝐷𝑒 Equivalent diameter, [m]

𝐴_𝑓 Flow area, [𝑚2_]

P Cross section perimeter, [m]

𝑚 Mass flow rate, [𝑘𝑔 𝑠 ]

Greek Symbols

𝜌 Density, [𝑘𝑔 𝑚3_]

𝛼 Thermal diffusivity, [𝑚2_]_𝑠

𝜙 Nanoparticle volume fraction 𝜇_𝑓 Dynamic viscosity, [𝑁𝑠 𝑚2_]

𝜏_𝑥 Wall local shear stress, [𝑁 𝑚2_]

𝜏 Wall average shear stress, [𝑁 𝑚2_]

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Chapter 1 INTRODUCTION

1.1 Introduction to Nanofluids

Vast range of industrial processes deals with the transfer of heat energy. In almost all industrial equipments heat should be added, removed or transferred from one part to another and this is a prominent task in industry.

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small solid particles to fluid so that the new suspension has greater conductivity compared to pure fluid. The possibility of using suspensions with the size of solid particles in the range of 2 millimeters or micrometers has been observed by some researchers in the past and following drawbacks have been reported (S. K. Das, Choi, & Patel, 2006):

1. The bigger the size of solid particles, higher is the chance of sedimentation. When solid particles settle on the surface of their container, heat transfer decreases.

2. High velocity of solid particle-fluid suspensions over a surface decreases the rate of sedimentation but increases the chance of surface erosion.

3. Clogging is another disadvantage of these suspensions especially in the case of narrow passages and microchannels.

4. Pressure drop increases considerably compared to the case of pure fluid.

5. Eventually the increment of thermal conductivity with the enhancement of solid particle concentration (i.e., increasing particle concentration causes higher thermal conductivity of the suspension and subsequently increasing the abovementioned problems).

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A properly dispersed nanofluid suspension has the following advantages compared to the conventional micro-sized particle-fluid suspensions (S. K. Das et al., 2006):

1. Increased conduction heat transfer as a result of increased surface area of the particles. Particles with diameters less than 20 nm carry 20% of their atoms on their surface make them better medium for heat transfer. Besides the small size of these particles let them move faster in the suspension which brings about micro-convection of fluid and hence increased heat transfer. This micro-convection in turn speeds up the dispersion of heat in the fluid. This is the main reason for increase in thermal conductivity of nanofluid by an increase in temperature.

2. Because of the small weight of particles the chances of sedimentation are less so the Brownian motion dominates their weight and as a result of that the suspension will be more stable.

3. Using micro-particles in microchannels always have the problem of clogging. The nanofluid overcomes this problem as it contains finer particles.

4. The small nanoparticles impart smaller momentum to solid walls of equipments such as exchangers, pipes, and pumps.

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thermal conductivity increase in nanofluid as far as there is not a sharp increase in nanofluid‟s viscosity.

Nanoparticles can be mainly divided into three categories: ceramic particles, pure metallic particles, and carbon nanotubes (CNTs) and some of the base fluids have been used so far include water, ethylene glycol, transformer oil, and toluene.

Models initially proposed for prediction of effective thermal conductivity of nanofluids (e.g. Maxwell-Garnett model (Garnett, 1904) and Hamilton-Crosser model (Hamilton & Crosser, 1962)) mainly included thermal conductivity of base fluid and particles, solid volume fraction of particles, and the shape of nanoparticles. Further experiments to measure nanofluids thermal conductivity proved an important dependence on nanoparticle size and temperature. Even small temperature change which does not affect the thermal conductivity of base fluid and nanoparticle, has a considerable effect on nanofluid thermal conductivity. This fact indicated that some kind of particle movement that dramatically changes with temperature must be taking place within the fluid.

While applying nanofluids for commercial cooling Tzeng et al. (Tzeng, Lin, & Huang, 2005) investigated the performance of both CuO(4.4% wt) and Al2O3(4.4%

wt) nanoparticles dispersed in automatic transmission oil as engine coolant. Comparing the results with conventional antifoam-oil coolant indicated that CuO nanofluid had the best heat transfer effect and antifoam-oil showed the worst effect.

1.2 Objective of Thesis

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average Nusselt numbers have been calculated numerically to measure the heat transfer rate of the flow.

Moreover, four different boundary conditions have been considered in the work as following:

1- Constant Wall Temperature and Uniform Entrance Velocity

2- Constant Wall Temperature and Parabolic Entrance Velocity

3- Constant Wall Heat Flux and Uniform Entrance Velocity

4- Constant Wall Heat Flux and Parabolic Entrance Velocity

Local and average Nusselt numbers have been calculated for all cases to find the effect of abovementioned boundary conditions on heat transfer rate of the channel.

1.3 Overview of Thesis Work

In chapter one an introduction to nanofluid and its history is briefly presented.

In chapter two a review over the related literature has been done and the models for nanofluid thermophysical properties are discussed to choose the best models among existing ones.

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chapter. An explanation about the numerical approach is presented at the end of this chapter.

In chapter four results in the form of graphs and tables are presented and discussed.

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Chapter 2 LITERATURE SURVEY

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Convective heat transfer plays an important role in a variety of thermal systems such as power plants, refrigerators, and small electronic devices. Using nanofluids in such equipments are helpful to increase the performance of cooling systems as well as reducing their size. Emerge of nanofluids opened a new way in heat transfer industry that can be the most impressive recent innovation in thermal science.

The idea of using solid-liquid mixtures is not a new idea, but in conventional mixtures millimeter or micrometer-sized particles were used, which had the disadvantages of clogging, erosion, sedimentation, and severe pressure drop. Recent advancements in material technology, during last decades has made it possible to make nano-size particles which can be used in solid liquid mixtures with an advantage of improved thermal properties and very small or none of the problems of conventional mixtures.

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Brownian motion rises with temperature, the temperature effect is considered which results in higher convection. This model is applicable to low concentration of solid particles. Moreover, this model considers the effect of the size of nanoparticle through an increase in specific surface of nanoparticles (S. Das, Sundararajan, Pradeep, & Patel, 2005). Likewise, an empirical constant 'c', links the temperature dependence of effective thermal conductivity to the Brownian motion of the particles. This constant (c) can be found by comparing the calculated value with experimental data, which comes in the order of 104. This empirical constant is adjustable and can be thought as a function of particle properties as well as size (Patel et al., 2005).

The same problem exists for effective viscosity of nanofluid. Comparing different correlations for nanofluids effective viscosity with experimental values from previous investigations on Al2O3nanofluids by Lee et al. (J.H. Lee, 2005) and Wang

et al. (X. Wang, Xu, & S. Choi, 1999) do not show good agreement. Einstein's model (Einstein, 1956) Brinkman's model (Brinkman, 1952), and Brownian motion effect's model underestimates effective viscosity of water based Al2O3 nanofluid compared

to mentioned experimental results, while Pak and Cho's correlation overestimates it in volume fractions more than 0.1 % (Hwang, Lee, & Jang, 2007). Based upon this comparison (Hwang et al., 2007) for volume fractions greater than 0.1%, Pak and Cho's model is not applicable, while Brinkman's model is more consistent with experimental results.

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enhancement of convective heat transfer coefficient with flow velocity as well as solid volume fraction (ϕ). They observed 39% increase in Nusselt number while increasing cupper particle concentration from 0% to 2% at constant Reynolds number. An experimental research on graphite-oil nanofluid by Yang et al. (Yang, Zhang, Grulke, Anderson, & Wu, 2005) for laminar flow in a horizontal tube heat exchanger revealed increase in static thermal conductivity. The enhancement of heat transfer coefficient was less than predicted by conventional correlations. Heris et al. (Heris, Esfahany, & Etemad, 2007) studied laminar flow, forced convection of Al₂O₃-water nanofluid experimentally in a circular tube with constant wall temperature. The thermal conductivity of nanofluid was calculated using the renovated Maxwell model (W. Yu & Choi, 2003) with liquid layer thickness as 10% of Al₂O₃ particle radius. It was observed that heat transfer increased with increase of Peclet number as well as ϕ. The heat transfer coefficient increase was much higher than predicted by heat transfer correlation applicable to the single phase fluid with nanofluid properties. A numerical investigation on γAl₂O₃-water nanofluid has been conducted by Roy et al. (Roy, Nguyen, & Lajoie, 2004) in a radial flow cooling system. They have observed 100% increase in heat transfer by 10 % increase in nanoparticle volume fraction. γAl₂O₃ is cubic aluminum oxide particle. They also found that wall shear stress increases as ϕ increases. Maiga et al. (Maiga, Nguyen, Galanis, & Roy, 2004) have studied laminar flow of γAl2O3-water and γAl2O3

-ethylene glycol nanofluids numerically in a tube with constant wall heat flux. They observed noticeable increase of heat transfer with increase in ϕ. The rate the heat transfer increase was higher for γAl2O3-EG. Wall shear stress also increased with ϕ.

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Chapter 3 MATHEMATICAL MODELING FOR TWO PHASE

FLOW

3.1 Introduction

In this chapter the governing continuity, momentum, and energy equations for a two-dimensional parallel plate channel are presented. Corresponding thermal conductivity and viscosity models for copper-water nanofluid is presented. Two Nusselt numbers are defined based on inlet temperature and nanofluid bulk temperature. The boundary conditions at top and bottom walls, channel entrance, and channel outlet are given in this chapter. A brief explanation about the numerical approach, momentum interpolation method, and the discretization scheme is presented in the last part of this chapter.

3.2 Geometry and Problem Statement

Figure 3.1 displays the problem geometry which is a rectangular duct of height

H=1cm, and length L=1m with very large depth compared to the height. „u‟ and „v‟

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Figure 3.1: Channel Geometry

Copper water is going to be used as nanofluid with spherical particles of uniform shape and size. The nanoparticle diameter is taken to be 100nm. The fluid flow is incompressible, steady state, and laminar and the nanofluid is assumed to be Newtonian. Particle distribution is homogeneous and solid liquid particles are in thermal equilibrium and flow at the same velocity. The hydrodynamic and thermophysical properties of particles and fluid is assumed to be constant and the values are considered at a fixed temperature of 0°C which are presented in Table 3.1. Buoyancy force is neglected because of small effect compared to flow, so the problem is a pure forced convection problem.

Table 3.1: Thermophysical properties of water and Cu at 0°C ((Jiji, 2006), (Incropera, Lavine, & DeWitt, 2011))

𝜌(𝑘𝑔/𝑚3₎ _𝐶

𝑝(𝐽/𝑘𝑔 𝐾) k (W/m K) 𝜇 (𝑘𝑔/𝑚 𝑠)

Pure water 999.8 4218 0.5619 1.791 × 10−3

Cu 8933 371 406

3.3 Governing Equations

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effect has been neglected as a result of high flow velocity. The general governing equations are as follow:

Continuity: 𝜕𝜌𝑛𝑓 𝜕𝑡 + 𝑑𝑖𝑣 𝜌𝑛𝑓𝑉 = 0 3.1 Momentum-X: 𝜕(𝜌𝑛𝑓𝑢) 𝜕𝑡 + 𝑑𝑖𝑣 𝜌𝑛𝑓𝑢𝑉 = − 𝜕𝑃 𝜕𝑥+ 𝑑𝑖𝑣(𝜇𝑛𝑓𝑔𝑟𝑎𝑑𝑢) 3.2 Momentum-Y: 𝜕(𝜌_𝑛𝑓𝑣) 𝜕𝑡 + 𝑑𝑖𝑣 𝜌𝑛𝑓𝑣𝑉 = − 𝜕𝑃 𝜕𝑦+ 𝑑𝑖𝑣(𝜇𝑛𝑓𝑔𝑟𝑎𝑑𝑣) 3.3 Energy: 𝜕(𝜌_𝑛𝑓𝐶_{𝑝 𝑛𝑓}𝑇) 𝜕𝑡 + 𝑑𝑖𝑣(𝜌𝑛𝑓𝐶𝑝 𝑛𝑓𝑉 𝑇) = 𝑑𝑖𝑣(𝑘𝑒𝑓𝑓𝑔𝑟𝑎𝑑𝑇) 3.4

Nanofluid density is obtained as follows:

𝜌𝑛𝑓 = 1 − 𝜙 𝜌𝑓+ 𝜙𝜌𝑝 3.5

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𝐶_{𝑝 𝑛𝑓} = 1 − 𝜙 𝜌𝑓𝐶𝑝 𝑓+ 𝜙𝜌𝑝𝐶𝑝 𝑝

𝜌_𝑛𝑓 3.6

Brinkman's model (Brinkman, 1952) is used for effective viscosity of nanofluid.

𝜇_𝑛𝑓 = 𝜇𝑓

(1 − 𝜙)2.5 3.7

Effective thermal conductivity of nanofluid is determined using Patel et al.'s model (Patel et al., 2005). For two-phase mixture the model gives

𝑘𝑒𝑓𝑓 𝑘_𝑓 = 1 + 𝑘𝑝𝐴𝑝 𝑘_𝑓𝐴_𝑓 + 𝑐𝑘𝑝𝑃𝑒 𝐴𝑝 𝑘_𝑓𝐴_𝑓 3.8

Here c is the only empirical constant of the model. Santra et al. (Santra, Sen, & Chakraborty, 2009) has found it for Cu-water nanofluid from the experimental data obtained by Xuan and Li (Xuan & Li, 2000). The average value of the constant is 3.64 × 104_.

Heat transfer area ratio of particle to fluid is calculated as

𝐴_𝑝 𝐴_𝑓 = 𝑑𝑓 𝑑_𝑝 𝜙 (1 − 𝜙) 3.9

Where 𝐴_𝑓 is the conduction heat transfer area of liquid medium and 𝐴_𝑝 is the corresponding area of solid particles. Here 𝑑𝑝is the solid particles diameter and 𝑑𝑓 is

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16 Peclet number is given as

𝑃𝑒 =𝑢𝑝𝑑𝑝

𝛼_𝑓 3.10

𝑢𝑝 is the Brownian motion velocity of the particles which is given by

𝑢_𝑝 = 2𝐾𝑏𝑇

𝜋𝜇_𝑓𝑑_𝑝2 3.11

where 𝐾_𝑏 is the Boltzmann constant (𝐾_𝑏 = 1.3807 × 10−23𝐽

𝐾 ).

Thermal diffusivity of nanofluid is

𝛼_𝑛𝑓 = 𝑘𝑒𝑓𝑓

𝜌_𝑛𝑓𝐶_{𝑝 𝑛𝑓} 3.12

Local and average Nusselt number at bottom hot wall have been calculated using two different definitions based on the difference between the wall and bulk temperatures. The definitions for both cases of constant wall temperature and constant wall heat flux are given in the following sections.

3.3.1 Nusselt Number for Constant Wall Temperature 3.3.1.1 Nu Based on 𝑻_𝒊𝒏:

Heat flux at the wall boundary is

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17 → 𝑕 = −𝑘𝑒𝑓𝑓

𝜕𝑇 𝜕𝑦 _{𝑦 =0}

𝑇𝑤−𝑇𝑖𝑛 (a)

Nusselt number for nanofluids is defined in terms of the conductivity of pure fluid as:

𝑁𝑢 =𝑕 𝐿𝑟𝑒𝑓

𝑘𝑓 (b)

Substituting h from (a) to (b) we have

𝑁𝑢_𝑥 = −𝐿_𝑟𝑒𝑓 𝑘𝑒𝑓𝑓 𝑘_𝑓

𝜕𝑇 𝜕𝑦 _𝑦=0

(𝑇_𝑤 − 𝑇_𝑖𝑛) 3.13

Where, 𝐿𝑟𝑒𝑓 is the reference length and is equal to the width of channel (h) and 𝑘𝑒𝑓𝑓

is a function of x.

The average Nusselt number will be:

𝑁𝑢 =1 𝐿 𝑁𝑢𝑥 𝑑𝑥 𝐿 0 3.14 3.3.1.2 Nu Based on 𝑻𝒃 𝑁𝑢_𝑥 = −𝐿_𝑟𝑒𝑓 𝑘𝑒𝑓𝑓 𝑘_𝑓 𝜕𝑇 𝜕𝑦 _𝑦=0 (𝑇_𝑤 − 𝑇_𝑏) 3.15

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18

𝑇𝑏 =

𝜌_𝐴_𝑐 𝑛𝑓𝐶_{𝑝 𝑛𝑓} 𝑢 𝑇 𝑑𝐴𝑐

𝜌_𝐴_𝑐 𝑛𝑓𝐶_{𝑝 𝑛𝑓} 𝑢 𝑑𝐴𝑐 3.16

In this case the average Nusselt number will be a function of x as well:

𝑁𝑢 =1

𝑥 𝑁𝑢𝑥 𝑑𝑥

𝑥

0 3.17

3.3.2 Nusselt Number for Constant Wall Heat Flux 3.3.2.1Nu Based on 𝑻𝒊𝒏 𝑁𝑢𝑥 = −𝑞𝑤𝐿𝑟𝑒𝑓 𝑘_𝑓(𝑇_𝑤− 𝑇_𝑖𝑛) 3.18 𝑁𝑢 =1 𝐿 𝑁𝑢𝑥 𝑑𝑥 𝐿 0 3.19 3.3.2.2 Nu Based on 𝑻_𝒃 𝑁𝑢_𝑥 = −𝑞𝑤𝐿𝑟𝑒𝑓 𝑘𝑓(𝑇𝑤− 𝑇𝑏) 3.20 𝑁𝑢 =1 𝐿 𝑁𝑢𝑥 𝑑𝑥 𝐿 0 3.21

Shear stress at the bottom wall is:

𝜏_𝑥 = 𝜇_𝑛𝑓 𝜕𝑢

𝜕𝑦 _𝑦=0 3.22

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19 𝜏 =1 𝐿 𝜏𝑥 𝑑𝑥 𝐿 0 3.23

Reynolds number is defined by:

𝑅𝑒 =𝜌𝑓 𝑈𝑟𝑒𝑓 𝐻

𝜇𝑓 3.24

Where H is channel width and 𝑈_𝑟𝑒𝑓 = 𝑈_𝑚 is the mean inlet velocity.

and 𝑅𝑒_𝐷_𝑒 as following:

𝑅𝑒_𝐷_𝑒 =𝜌𝑓 𝑈𝑟𝑒𝑓 𝐷𝑒

𝜇_𝑓 3.25

where 𝐷_𝑒 (equivalent diameter) is given by:

𝐷_𝑒 =4 𝐴𝑓

𝑃 = 2H 3.26

Using equations 3.26, 3.27, and 3.28 we have:

𝑅𝑒_𝐷_𝑒 = 2 𝑅𝑒 3.27

3.4 Boundary Conditions

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20

For the energy equation, flow enters the channel at a low uniform temperature and leaves it while it is thermally developed. Wall boundary conditions are constant temperature and constant heat flux cases.

3.4.1 Entrance Boundary Conditions

Uniform entrance velocity:

At x=0, 0 ≤ y ≤ H: 𝑢 = 𝑢_𝑚, v=0, 𝑇 = 𝑇_𝑐 Where 𝑢_𝑚 is the mean velocity at the inlet. with 𝑢𝑚 = 1 𝑚/𝑠 and 𝑇𝑐 = 0℃

Parabolic entrance velocity:

At x=0, 0 ≤ y ≤ H: 𝑢 𝑦 =−6𝑢𝑚

𝐻2 𝑦2+

6𝑢𝑚

𝐻 𝑦 , v=0, 𝑇 = 𝑇𝑐

with 𝑢𝑚 = 1 𝑚/𝑠 and 𝑇𝑐 = 0℃

3.4.2 Outlet Boundary Conditions

For all variables zero gradient condition have been considered at the outlet: At x=L, 0 ≤ y ≤ H: 𝜕𝑢_𝜕𝑥 = 0, 𝜕𝑣_𝜕𝑥 = 0, 𝜕𝑇_𝜕𝑥 = 0

and to ensure overall mass conservation, correcting the axial velocity component at the outlet is necessary:

𝑢𝑁,𝑗 = 𝑢𝑁−1,𝑗

𝑚 𝑖𝑛

𝑚 𝑜𝑢𝑡

where N refers to a point at the outlet boundary, and

𝑚 _𝑖𝑛 = 𝜌_𝐴 𝑛𝑓 𝑢𝑖𝑛𝑑𝐴 and 𝑚 𝑜𝑢𝑡 = 𝜌_𝐴 𝑛𝑓 𝑢𝑁,𝑗𝑑𝐴

3.4.3 Wall Boundary Conditions

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21 At y=0, 0 ≤ x ≤ L: u=0, v=0, 𝑇 = 𝑇_𝑕

At y=H, 0 ≤ x ≤ L: u=0, v=0, 𝑇 = 𝑇_𝑕

with 𝑇_𝑕 = 1℃

Constant wall heat flux:

At y=0, 0 ≤ x ≤ L: u=0, v=0, 𝑞 = 𝑞_𝑤

At y=H, 0 ≤ x ≤ L: u=0, v=0, 𝑞 = 𝑞_𝑤

with 𝑞_𝑤 _{= 10 𝑊 𝑚}2

3.5 Numerical Method

A Fortran code has been developed to find the velocity and temperature profile of a flow in a rectangular channel.

Mass, momentum, and energy conservation equations have been discretized by a control volume approach. The general form of differential equations for a steady state fluid flow is as following:

𝜕(𝜌𝑛𝑓𝑢𝜑 ) 𝜕𝑥 + 𝜕(𝜌𝑛𝑓𝑣𝜑) 𝜕𝑦 = 𝜕 𝜕𝑥 Г 𝜕𝜑 𝜕𝑥 + 𝜕 𝜕𝑦 Г 𝜕𝜑 𝜕𝑦 + 𝑆𝜑

Convection term Diffusion term Source term

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22

With 𝜑 = 1 and Г = 0 in continuity equation, 𝜑 = 𝑢 or 𝑣 and Г = 𝜇_𝑛𝑓 in momentum conservation equation, and 𝜑 = 𝑇 and Г = 𝑘𝑒𝑓𝑓

𝐶_{𝑝 𝑛𝑓} in energy equation.

The diffusion term is discretized using central difference scheme, while CUBISTA scheme (Convergent and Universally Bounded Interpolation Scheme for Treatment of Advection) (Alves, Oliveira, & Pinho, 2003) is used to discretize the convection term. This is a TVD (total-variation diminishing) high resolution scheme (HRS) with third order accuracy. The face value for uniform meshes in this scheme is given by:

7₄𝜑 𝑃 0 < 𝜑 𝑃 <3₈

𝜑_𝑓= 3₄𝜑 _𝑃 +3₈ 3₈ ≤ 𝜑 _𝑃 ≤3₄ 3.29

1₄𝜑 _𝑃+3₄ 3₄< 𝜑 _𝑃 < 1 𝜑 _𝑃 elsewhere

Transforming the relations for the case of non-uniform meshes gives:

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23 𝑋 𝑃 = 𝑋_𝑋𝑃−𝑋𝑈 𝐷−𝑋𝑈 , 𝑋 𝑓 = 𝑋𝑓−𝑋𝑈 𝑋𝐷−𝑋𝑈 and 𝜑 𝑃 =𝜑_𝜑𝑃−𝜑𝑈 𝐷−𝜑𝑈 , 𝜑 𝑓 = 𝜑𝑓−𝜑𝑈 𝜑𝐷−𝜑𝑈

Here X stands for x or y coordinates and the subscripts U and D refer to upstream and downstream cells to cell P which is, itself, upstream of the cell face f under consideration.

Since non-staggered (collocated) grid system has been employed the problem of checker board pressure field has been avoided by using momentum interpolation method (B. Yu et al., 2002), and to find the pressure profile the SIMPLE algorithm has been employed.

Figure 3.2: Nonstaggered grid arrangement

Discretizing any of continuity, momentum, or energy equations with general variable

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25

Here 𝜑_𝑒, 𝜑_𝑤, 𝜑_𝑛, and 𝜑_𝑠 are being calculated using abovementioned CUBISTA method and 𝑢_𝑒, 𝑢_𝑤, 𝑣_𝑛, and 𝑣_𝑠 are being found using momentum interpolation method (B. Yu et al., 2002). The following equation shows the calculation of 𝑢_𝑒 (similar relations are being used for other faces).

[𝑓_𝑒+_(𝐴 𝑃)𝐸𝑢𝐸+ (1 − 𝑓𝑒+)(𝐴𝑃)𝑃𝑢𝑃] +𝛼𝑢[(𝑓𝑒+ 𝑆𝑐 𝐸+ (1 − 𝑓𝑒+)(𝑆𝑐)𝑃)𝛿𝑥𝑒∆𝑦 −𝑓𝑒+(𝑆𝑐)𝐸∆𝑥𝐸∆𝑦 − (1 − 𝑓𝑒+)(𝑆𝑐)𝑃∆𝑥𝑃∆𝑦] 3.36 𝑢_𝑒 = _(𝐴1 𝑃)𝑒 +𝛼𝑢[−∆𝑦 𝑃𝐸 − 𝑃𝑃 + 𝑓𝑒 +_∆𝑦(𝑃 𝑒 − 𝑃𝑤)𝐸 +(1 − 𝑓_𝑒+_{)∆𝑦(𝑃} 𝑒 − 𝑃𝑤)𝑃] +(1 − 𝛼_𝑢)[𝑢_𝑒0_𝐴 𝑃 𝑒 − 𝑓𝑒+𝑢𝐸0 𝐴𝑃 𝐸− (1 − 𝑓𝑒+)𝑢𝑃0(𝐴𝑃)𝑃]

where (𝐴𝑃)𝑒 is being interpolated as:

(𝐴_𝑃)_𝑒 = 𝑓_𝑒+_{( 𝐴} 𝑖 𝑖 )𝐸+ 1 − 𝑓𝑒+ 𝐴𝑖 𝑖 𝑃− [𝑓𝑒+ 𝑆𝑃 𝐸+ (1 − 𝑓𝑒+)(𝑆𝑃)𝑃]𝛿𝑥𝑒∆𝑦 3.37 and 𝑓𝑒+=_2𝛿𝑥∆𝑥 𝑒

Here superscript 0 refers to previous iteration and subscripts e and w refer to east and west faces respectively and subscript E refers to east cell. 𝛼𝑢 stands for axial velocity

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26

Chapter 4 RESULTS AND DISCUSSION

4.1 Introduction

The heat transfer problem associated with the flow between two parallel plates has been studied numerically for a two phase flow with symmetric top and bottom thermal boundary conditions. The geometry of the problem is presented in Fig. 3.1. The channel is 1 m in length while the height is 1 cm. The nanofluid is composed of water as base fluid with copper particles of 100 nm diameter suspended in it. The flow and temperature fields are studied for a range of Re and ϕ. Thermal conductivity of nanofluid has been calculated using Patel et. al.‟s correlation for each control volume as it depends on temperature. The constant „c‟ in the correlation has been calculated from experimental results for copper-water nanofluid (Xuan & Li, 2000). The average value of constant is used in our work, which is 3.60 × 104. Thermophysical properties of water and copper at the base temperature, i.e. at 0℃ are summarized in Table 3.1. Results are presented and compared for both cases of uniform and parabolic entrance velocity while the walls are kept at constant temperature and constant heat flux. Re changes from 10 to 1500 while ϕ has been varied from 0 to 5 %.

4.2 Grid Independence Study

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27

entrance velocity and walls at constant heat flux. It is found that 201 grids along X-direction by 41 grids along Y-X-direction give satisfactory results. Further increase in the number of grids does not affect the results noticeably. The results of the grid independence study have been summarized in Table 4.1.

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Table 4.1: Results of grid independence test, at Re=500, pure fluid

No. of grids in X-direction

No. of grids in Y-direction

Average Nu number at the bottom wall

Average wall shear stress 201 21 5.845399 𝟑. 𝟐𝟏𝟐𝟗𝟐𝟔𝟐 × 𝟏𝟎−𝟐 301 21 5.843040 𝟑. 𝟐𝟏𝟑𝟔𝟒𝟒𝟒 × 𝟏𝟎−𝟐 401 21 5.842008 𝟑. 𝟐𝟏𝟑𝟔𝟑𝟔𝟔 × 𝟏𝟎−𝟐 501 21 5.841102 𝟑. 𝟐𝟏𝟒𝟗𝟏𝟐𝟗 × 𝟏𝟎−𝟐 601 21 5.842097 𝟑. 𝟐𝟏𝟎𝟎𝟒𝟒𝟕 × 𝟏𝟎−𝟐 201 31 5.923468 𝟑. 𝟐𝟑𝟕𝟐𝟒𝟗𝟕 × 𝟏𝟎−𝟐 301 31 5.920895 𝟑. 𝟐𝟑𝟗𝟑𝟖𝟏𝟕 × 𝟏𝟎−𝟐 401 31 5.919976 𝟑. 𝟐𝟑𝟗𝟏𝟓𝟓𝟔 × 𝟏𝟎−𝟐 501 31 5.920148 𝟑. 𝟐𝟒𝟏𝟑𝟗𝟏𝟗 × 𝟏𝟎−𝟐 601 31 5.919202 𝟑. 𝟐𝟒𝟐𝟐𝟗𝟑𝟎 × 𝟏𝟎−𝟐 201 41 5.954765 𝟑. 𝟐𝟒𝟔𝟑𝟔𝟑𝟔 × 𝟏𝟎−𝟐 301 41 5.952280 𝟑. 𝟐𝟓𝟎𝟐𝟖𝟏𝟗 × 𝟏𝟎−𝟐 401 41 5.951772 𝟑. 𝟐𝟓𝟑𝟑𝟖𝟗𝟓 × 𝟏𝟎−𝟐 201 81 5.986251 𝟑. 𝟐𝟓𝟕𝟒𝟐𝟑𝟔 × 𝟏𝟎−𝟐 401 81 5.985945 𝟑. 𝟐𝟔𝟏𝟔𝟐𝟕𝟏 × 𝟏𝟎−𝟐

4.3 Validation of Code

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𝑃𝑟 =𝜇𝑓𝐶𝑝 𝑓

𝑘_𝑓 4.1

𝜁 = 2 𝑥

𝐻 𝑅𝑒 5.1

The results for two cases of constant wall temperature and constant wall heat flux have been summarized in Table 4.2 and Table 4.3 respectively. The percentage deviations from theoretical values are presented for each case.

4.3.1 Constant Wall Temperature

Table 4.2: Local wall Nusselt number for clear fluid, constant wall temperature, comparison with theoretical results

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4.3.2 Constant Wall Heat Flux

Table 4.3: Local wall Nusselt number for clear fluid, constant wall heat flux, comparison with theoretical results

𝑵𝒖𝑿 , Pr=0.1 𝑵𝒖𝑿 , Pr=1.0 𝑵𝒖𝑿 , Pr=10 𝜻 (R. Das & Mohanty, 1983) Present Study Deviation (%) (R. Das & Mohanty, 1983) Present Study Deviation (%) (R. Das & Mohanty, 1983) Present Study Deviation (%) 0.002 6.893 8.172 18.555 15.330 16.916 10.345 31.740 35.639 12.284 0.0085 4.379 5.064 15.643 8.378 8.983 7.221 17.471 18.641 6.697 0.020 4.242 4.443 4.738 5.912 6.487 9.726 12.058 13.051 8.235 0.057 4.156 4.181 0.602 4.352 4.752 9.189 8.248 8.767 6.296 0.071 4.143 4.153 0.241 4.146 4.539 9.490 7.647 8.125 6.249 0.660 4.117 4.113 -0.097 4.158 4.525 8.826

It has been observed that the percentage deviation from analytical solution for channel with constant wall temperature remains below 10.015% while 18.56 % deviation has been observed for channel with constant heat flux for small values of

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4.4 Effect of Various Parameters on Local and Average Nusselt

Numbers (Based on

𝑻

_𝒊𝒏

and

𝑻

_𝒃

)

4.4.1 Effect of Solid Volume Fraction and Reynolds Number on Local Wall Nusselt Number (Based on 𝑻_𝒊𝒏)

4.4.1.1 Constant Wall Temperature, and Uniform Entrance Velocity

The local Nusselt number based on 𝑇_𝑖𝑛 is shown in Fig. 4.1 and Fig. 4.2 at constant wall temperature and uniform inlet velocity for Re=100 and Re=1500 respectively.

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32

Figure 4.2: Distribution of local Nusselt number at hot wall for different solid volume fractions, Re=1500

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Figure 4.3 depicts the local Nusselt number based on 𝑇_𝑖𝑛 at constant wall temperature and uniform inlet velocity for ϕ=2.5%.

Figure 4.3: Distribution of local Nusselt number at hot wall for different Reynolds numbers, ϕ=2.5%.

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4.4.1.2 Constant Wall Temperature, Comparison Between, Uniform and Parabolic Entrance Velocities

The local Nusselt number based on 𝑇𝑖𝑛 is shown in Fig. 4.4 and Fig. 4.5 at constant

wall temperature for uniform and parabolic inlet velocities for Re=100 and Re=1000 respectively.

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35

Figure 4.5: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different solid volume fractions, Re=1000

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36

Figure 4.6 depicts the local Nusselt number based on 𝑇_𝑖𝑛 at constant wall temperature, for uniform and parabolic inlet velocities for ϕ=2.5%.

Figure 4.6: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different Reynolds numbers, ϕ=2.5%

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4.4.1.3 Constant Wall Heat Flux, and Uniform Entrance Velocity

The local Nusselt number based on 𝑇_𝑖𝑛 is shown in Fig. 4.7 and Fig. 4.8 at constant wall heat flux and uniform inlet velocity for Re=100 and Re=1500 respectively.

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38

Figure 4.8: Distribution of local Nusselt number at hot wall for different solid volume fractions, Re=1500

As indicated in Fig. 4.7 the local wall Nusselt number based on 𝑇𝑖𝑛 also decreases

with x for the case of walls with constant heat flux. Increasing wall temperature with

x means higher difference between wall and inlet temperature, which causes the local

Nusselt number to fall (Eq. 3.18). At higher values of solid particle volume fraction higher conductivity of nanofluid increases the rate of heat transfer between the walls and nanofluid which keeps the wall temperature lower and results in higher values of local Nusselt number.

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39

Figure 4.9: Distribution of local Nusselt number at hot wall for different Reynolds numbers, ϕ=2.5%.

As Reynolds number increases higher rate of heat transfer causes the wall temperature to remain lower which enhances the local wall Nusselt number.

4.4.1.4 Constant Wall Heat Flux, Comparison Between, Uniform and Parabolic Entrance Velocities

The local Nusselt number based on 𝑇_𝑖𝑛 is shown in Fig. 4.10 and Fig. 4.11 at constant wall heat flux for uniform and parabolic inlet velocities for Re=100 and

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40

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41

Figure 4.11: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different solid volume fractions, Re=1000

The same course of changes as channel with uniform entrance velocity is observed for channels at constant heat flux in Fig 4.10 and Fig. 4.11. The local Nusselt decreases with x and increases with ϕ.

As depicted in the figure, the channel with uniform entrance velocity, shows higher values of local Nusselt number close to the flow entrance region but, for all values of solid volume fraction, this difference decreases with x until the Nusselt number values for channel with parabolic entrance velocity exceeds the Nusselt values for channel with uniform entrance velocity.

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42

Figure 4.12: Comparison of local Nusselt number along hot wall between uniform (U) and parabolic (P) entrance velocities, different Reynolds numbers, ϕ=2.5%.

In both cases of uniform and parabolic entrance velocities, the local Nusselt number increases with Reynolds number, at a constant value of solid volume fraction.

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4.4.2 Average Wall Nusselt Number (Based On 𝑻_𝒊𝒏) 4.4.2.1 Constant Wall Temperature

The variation of average Nusselt number with solid volume fraction (ϕ) for different

Re is presented in Fig. 4.13 for the cases of uniform and parabolic entrance velocities. Here average Nusselt number is calculated based on 𝑇_𝑖𝑛 (Eq. 3.14), between x=0 and x=L where L= 1m. For all cases the flow is hydrodynamically fully developed at the exit.

Figure 4.13: Average Nusselt number at hot wall for different Re and ϕ for uniform (U) and parabolic (P) entrance velocities, constant wall temperature

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44

velocity changes from parabolic to uniform. On the other hand this change is 4.4% if a pure fluid is used. This increase in 𝑁𝑢 for a channel with uniform inlet flow can be interpreted as a result of hydrodynamic entrance region with high local Nu due to developing velocity boundary layer, which is desirable in practical heat transfer applications.

Figure 4.14 illustrates the augmentation of Nusselt ratio of nanofluid to pure fluid (𝑁𝑢 𝑁𝑢𝑛𝑓 𝑓) as a function of solid volume fraction (ϕ) for different Reynolds

numbers. The results for both cases of uniform and parabolic entrance velocities have been depicted in the figure. In both cases the enhancement of nanofluid Nusselt number is much larger for higher values of Re. That is because, for lower Re due to small flow momentum, the temperature rapidly reaches the wall temperature regardless of the volume fraction of nanoparticles. However at higher Re a continuous increase of Nusselt ratio is evident as ϕ increases. This is due to increase of nanofluid conductivity as a result of increase in solid volume fraction. The Nusselt ratio increase for uniform inlet velocity is smaller than the increase for parabolic inlet velocity at Re=10 while, it is vice versa for Re=1500.

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45

Figure 4.14: uniform (U) and parabolic (P) entrance velocities, L=1m, 𝑻𝒘=constant

Table 4.4: Effect of solid volume fraction on the average Nu number (𝑵𝒖 ), uniform entrance velocity 𝜙 = 0% 𝜙 = 1% 𝜙 = 2% 𝜙 = 3% 𝜙 = 4% 𝜙 = 5% Re=10 0.380 0.380 0.380 0.380 0.381 0.381 Re=100 2.426 2.530 2.624 2.710 2.789 2.860 Re=500 4.671 4.952 5.225 5.504 5.773 6.048 Re=1500 7.002 7.441 7.879 8.281 8.704 9.123

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4.4.2.2 Constant Wall Heat Flux

The variation of average Nusselt number with solid volume fraction (ϕ) for different

Re is presented in Fig. 4.15 for the cases of uniform and parabolic entrance velocities. The same result has been presented for channel with constant temperature walls. Here average Nusselt number is calculated based on 𝑇_𝑖𝑛 (Eq. 3.19), between

x=0 and x=L where L= 1m. For all cases the flow is hydrodynamically fully

developed at the exit.

Figure 4.15: Average Nusselt number at hot wall for different Re and ϕ for uniform (U) and parabolic (P) entrance velocities, constant wall heat flux

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47 𝑁𝑢

increases by 11.7% for pure fluid and 13% for nanofluid with ϕ=5% when entrance velocity changes from parabolic to uniform.

The augmentation of Nusselt ratio (𝑁𝑢 𝑁𝑢_𝑛𝑓 _𝑓) with solid volume fraction (ϕ) is presented in Fig. 4.16 for different Reynolds numbers and for constant wall heat flux. As shown in the figure, the results at smaller values of Re overlap, while at higher Re more increase of Nu ratio is observed for channel with uniform entrance velocity as solid volume fraction increases.

In Table 4.6 and Table 4.7 values are given for channels with uniform and parabolic entrance velocities. Using the values at Re=1500 we come across to a 31.3% and a 30.0% increase of 𝑁𝑢 respectively for uniform and parabolic inlet velocities as a result of 5% increase in nanoparticle volume fraction.

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Table 4.6: Effect of solid volume fraction on the average Nusselt number (𝑵𝒖 ), uniform entrance velocity

𝜙 = 0% 𝜙 = 1% 𝜙 = 2% 𝜙 = 3% 𝜙 = 4% 𝜙 = 5%

Re=10 0.960 0.990 1.018 1.045 1.070 1.094

Re=100 3.238 3.417 3.591 3.760 3.924 4.084

Re=500 5.964 6.332 6.696 7.056 7.414 7.769

Re=1500 9.122 9.703 10.279 10.850 11.417 11.981

Table 4.7: Effect of solid volume fraction on the average Nusselt number (𝑵𝒖 ), parabolic entrance velocity

𝜙 = 0% 𝜙 = 1% 𝜙 = 2% 𝜙 = 3% 𝜙 = 4% 𝜙 = 5%

Re=10 0.946 0.975 1.003 1.029 1.053 1.077

Re=100 3.147 3.320 3.487 3.650 3.808 3.962

Re=500 5.609 5.947 6.280 6.610 6.938 7.263

Re=1500 8.169 8.664 9.154 9.639 10.120 10.599

4.4.2.3 Comparison between Two Cases of Constant Wall Temperature and Constant Wall Heat Flux

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Figure 4.17: Average Nusselt number at hot wall, different Re and ϕ, for constant wall temperature (cons. T) and constant wall heat flux (cons. q), channel with

uniform entrance velocity

At all Re and ϕ values the 𝑁𝑢 is higher when walls are at constant heat flux. At

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Figure 4.18: walls at constant temperature and constant heat flux, uniform entrance velocity, L=1m

The increase of Nusselt ratio (𝑁𝑢 𝑁𝑢_𝑛𝑓 _𝑓) with solid volume fraction (ϕ) is presented in Fig. 4.18 for different Reynolds numbers and for uniform entrance velocity when walls are considered at both constant temperature and constant heat flux.

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In Fig. 4.19 the change of average Nusselt number ( based on 𝑇_𝑖𝑛) is shown with particle volume fraction. The results for two cases of walls at constant temperature and constant heat flux are compared while the entrance velocity profile is parabolic.

Figure 4.19: Average Nusselt number at hot wall, different Re and ϕ, for constant wall temperature (cons. T) and constant wall heat flux (cons. q), channel with

parabolic entrance velocity

Similar to the case of channel with uniform entrance velocity at all Re and ϕ values the 𝑁𝑢 is higher when walls are at constant heat flux. At ϕ=0% and Re=100 the 𝑁𝑢 increases by 29.7% when the wall changes from constant temperature to constant heat flux while at ϕ=0% and Re=1500 this increment will be 21.8%. Likewise at

ϕ=5% and Re=100 the 𝑁𝑢 increases by 38.3% and at Re=1500 by 22% when the wall changes from constant temperature to constant heat flux.

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Figure 4.20: walls at constant temperature and constant heat flux, parabolic entrance velocity, L=1m

When walls are at constant heat flux, more increase of Nusselt ratio with nanoparticle volume fraction is observed compared to the case of walls at constant temperature. This difference is higher at lower Re. At Re=100, 18.1% and 25.9% increase in 𝑁𝑢 is observed, respectively for walls at constant temperature and constant heat flux, as a result of 5% increase in nanoparticle volume fraction. While at Re=1500 these increments are 29.6% and 29.7%.

4.4.3 Local and Average Wall Nusselt Numbers (Based on 𝑻_𝒃)

Here local and average Nusselt numbers based on local bulk temperature 𝑇_𝑏 are given respectively by equations 3.15 and 3.17 for walls at constant temperature and by equations 3.20 and 3.21 for walls at constant heat flux. Results are presented for a pure fluid.

4.4.3.1 Constant Wall Temperature

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4.22, for uniform and parabolic inlet velocities respectively, for different values of Reynolds numbers.

Figure 4.21: Local and average Nusselt number vs 𝜉 = 𝑥/𝐷 𝑒

numbers, uniform entrance velocity for a pure fluid (ϕ=0.0%)

Figure 4.22: Local and average Nusselt number vs 𝝃 = 𝒙/𝑫 𝒆

𝑹𝒆_𝑫𝒆𝑷𝒓 for different Reynolds

numbers, parabolic entrance velocity for a pure fluid (ϕ=0.0%)

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Re. In both cases of uniform and parabolic entrance velocities the local and average

Nusselt numbers converge to 3.7 ( 𝑁𝑢 _𝜉→∞ = 𝑁𝑢 _𝜉→∞ = 3.7).

In the following pictures (Fig. 4.23 and Fig. 4.24) The local and average Nusselt numbers (based on 𝑇_𝑏) have been compared for two cases of uniform and parabolic entrance velocities while walls are at constant temperature for a pure fluid.

Figure 4.23: Local Nusselt number vs 𝝃 = 𝒙/𝑫 𝒆

𝑹𝒆_𝑫𝒆𝑷𝒓 for uniform and parabolic entrance

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Figure 4.24: Average Nusselt number vs 𝝃 = 𝒙/𝑫 𝒆

𝑹𝒆_𝑫𝒆𝑷𝒓 for uniform and parabolic

entrance velocities and different Reynolds numbers for a pure fluid (ϕ=0.0%)

Average Nusselt numbers (based on 𝑇_𝑏) with respect to 𝜉 for pure fluid flow in channel with walls at constant temperature are shown in Fig. 4.24, for uniform and parabolic inlet velocities, for different values of Reynolds numbers.

As seen in Fig. 4.23 and Fig. 4.24 the local and average Nusselt numbers are higher for channel with uniform entrance velocity, compared to the channel with parabolic entrance velocity at smaller values of 𝜉 while for higher values of 𝜉, they converge to the same value of 3.7.

Local and average Nusselt number ratios (based on 𝑇𝑏) of the nanofluid (with ϕ=5%)

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Figure 4.25: Local (Loc) and average (Ave) Nusselt number ratios, vs 𝝃 = 𝒙/𝑫 𝒆

𝑹𝒆_𝑫𝒆𝑷𝒓 for

uniform entrance velocity and constant wall temperature for ϕ=5.0%

parabolic entrance velocity and constant wall temperature for ϕ=5.0%

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parabolic entrance velocities. In the case of channel with parabolic entrance velocity scatter of data is seen at smaller 𝜉 when Reynolds number is small.

4.4.3.2 Constant Wall Heat Flux

Local and average Nusselt numbers (based on 𝑇_𝑏) with respect to 𝜉 for pure fluid flow in channel with walls at constant heat flux are shown in Fig. 4.27 and Fig. 4.28, respectively for uniform and parabolic inlet velocities, for different values of Reynolds numbers.

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numbers, parabolic entrance velocity for a pure fluid (ϕ=0.0%)

Similar to the case of walls at constant temperature, the local and average Nusselt numbers defined by Eq. 3.20 and 3.21 have to overlap for different values of Reynolds number, but as a result of numerical error as x approaches zero where Nusselt number approaches infinity, deviations are observed. For both uniform and parabolic entrance velocities the local and average Nusselt numbers converge to 4.1 ( 𝑁𝑢 𝜉→∞ = 𝑁𝑢 𝜉→∞ = 4.1).

In the following figures (Fig. 4.29 and Fig. 4.30) the local and average Nusselt numbers (based on 𝑇𝑏) have been compared for two cases of uniform and parabolic

entrance velocities while walls are at constant heat flux.

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𝑹𝒆_𝑫𝒆𝑷𝒓 for uniform and parabolic entrance

velocities and different Reynolds numbers for a pure fluid (ϕ=0.0%)

Local Nusselt numbers (based on 𝑇_𝑏) with respect to 𝜉 for pure fluid flow in channel with walls at constant heat flux are shown in Fig. 4.30, for uniform and parabolic inlet velocities, for different values of Reynolds numbers.

𝑹𝒆_𝑫𝒆𝑷𝒓 for uniform and parabolic

entrance velocities and different Reynolds numbers for a pure fluid (ϕ=0.0%)

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entrance velocity at smaller values of 𝜉 while for higher values of 𝜉, they converge to the same value (4.1).

Comparisons of local and average Nusselt numbers (based on 𝑇_𝑏) between channels with walls at constant temperature, and constant heat flux are depicted in Fig 4.31 and Fig. 4.32 respectively for the case of uniform entrance velocity.

Figure 4.31: Local Nusselt number vs 𝝃 = 𝒙/𝑫 𝒆

𝑹𝒆_𝑫𝒆𝑷𝒓 for constant wall temperature (C.T)

and constant wall heat flux (C.q), uniform entrance velocity and different Reynolds numbers for a pure fluid (ϕ=0.0%)

𝑹𝒆_𝑫𝒆𝑷𝒓 for constant wall temperature

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As shown in the figures both local and average Nusselt numbers are higher when walls are at constant heat flux compared to the channels with walls at constant temperature and as 𝜉 → ∞ local and average Nusselt numbers for the channel with walls at constant heat flux approach to 4.1 while for channel with walls at constant temperature they approach to 3.7.

The ratio of local (𝑁𝑢𝑛𝑓 𝑁𝑢𝑓) and average (𝑁𝑢 𝑁𝑢𝑛𝑓 𝑓) Nusselt numbers (based on

𝑇_𝑏) with 5% solid volume fraction nanofluid, are shown in Fig. 4.33 and Fig. 4.34 respectively for channel with uniform and parabolic inlet velocities, where walls are at constant heat flux.

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parabolic entrance velocity and constant wall heat flux for ϕ=5.0%

As seen in pictures, the ratio of local (𝑁𝑢_𝑛𝑓 𝑁𝑢_𝑓) and average (𝑁𝑢 𝑁𝑢_𝑛𝑓 _𝑓) Nusselt numbers increase as 𝜉 increases, and after a specific value of 𝜉, it remains constant at about 1.49 for local and 1.43 for average ratios, for both cases of uniform and parabolic entrance velocities. In Fig. 4.34 for the channel with parabolic entrance velocity scatter of data is observed at small values of x when Reynolds number is small.

4.5 Investigation of Solid Volume Fraction, Reynolds Number, and

Entrance Velocity Effects on Local and Average Shear Stresses

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4.5.1 Local Wall Shear Stress 4.5.1.1 Uniform Entrance Velocity

The local wall shear stress is shown in Fig. 4.35 and Fig. 4.36 for uniform inlet velocity, with Re=100 and Re=1500 respectively, for different values of solid volume fractions.

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Figure 4.36: Variation of local wall friction factor with x for different solid volume fractions for uniform inlet velocity, Re=1500.

As indicated in Fig. 4.35 and Fig. 4.36 the local wall shear stress increases with ϕ at a constant Reynolds number. As solid volume fraction increases, the dynamic viscosity of nanofluid (𝜇_𝑛𝑓) increases, which results in an enhancement of 𝜏_𝑥 (Eq. 3.22). Moreover the wall local shear stress decreases with x as velocity profile develops and remains constant when the flow is fully developed.

4.5.1.2 Parabolic Entrance Velocity

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Figure 4.37: Variation of local wall shear stress with x for different solid volume fractions for parabolic inlet velocity, Re=100

Figure 4.38: Variation of local wall shear stress with x for different solid volume fractions for parabolic inlet velocity, Re=1500

Similar to the uniform entrance velocity the local wall shear stress increases with solid volume fraction as the dynamic viscosity of nanofluid increases with ϕ at a constant Reynolds number. But unlike the last case the 𝜏𝑥 does not change with axial

distance. Since the velocity profile is developed at the entrance it does not change with x, which keeps a constant 𝜕𝑢_𝜕𝑦 along the x direction and results in a constant local shear stress at walls.

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The variation of average wall shear stress with solid volume fraction (ϕ) for different

Re is presented in Fig. 4.39 for the cases of uniform and parabolic entrance

velocities, where the ratio of length to the width of channel is _𝐻𝐿 = 100.

Figure 4.39: Average shear stress at walls for different Re and ϕ for uniform (U) and parabolic (P) entrance velocities

As illustrated in Fig. 4.39 average wall shear stress increases with increase in nanoparticle volume fraction for both cases. This happens as a result of nanofluid viscosity augmentation, with solid volume fraction. The augmentation of 𝜏 at Re=10 is 13.7% when ϕ is increased from 0% to 5%, while this augmentation is 17.5% at

Re=1500 (for channel with uniform entrance velocity). For parabolic entrance

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Chapter 5 CONCLUSION

Hydrodynamic and thermal behavior of laminar flow in a rectangular duct has been studied with copper-water nanofluid coolant, considered as a Newtonian fluid. The range of the Reynolds numbers considered is between 10 and 1500, while solid volume fraction is considered to change between 0% and 5%. The diameter of nanoparticle is assumed to be 100 nm and thermophysical properties of the base fluid and the nanoparticle is considered to be constant at the inlet temperature. To determine the effective viscosity of nanofluid brinkman (Brinkman, 1952) model has been used while the model proposed by Patel et al. (Patel et al., 2005) has been utilized to find the effective thermal conductivity.

Numerical Investigation of Copper-Water Nanofluid in a Parallel Plate Channel